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Chapter 6.
Descriptive Statistics
6.1
6.2
6.3
6.4
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Experimentation
Data Presentation
Sample Statistics
Examples
• Data: a mixture of nature and noise.
• Is the noise manageable?
 The noise is desired to be represented by a probability
distribution.
• Statistical inference:
– The science of deducing properties of an underlying
probability distribution from data
•
Can we have information on the underlying probability distribution?
 The information is given in the form of (functions of) data.
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Figure 6.1 The relationship between probability theory and statistical inference
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6.1 Experimentation
6.1.1 Samples
• Population: the set of all the possible observations available from a
particular probability distribution.
•
Sample: a subset of a population.
•
Random sample: a sample where the elements are chosen at random from
the population
•
A sample is desired to be representative of the population.
•
Types of observations: numerical and nominal
x
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6.1.2 Examples
• Example 1: Machine breakdowns
 Suppose that an engineer in charge of the maintenance of a
machine keeps records on the breakdown causes over a
period of a year.
 Suppose that 46 breakdowns were observed by the engineer
(see Figure 6.2).
 What is the population from which this sample is drawn?
 Factors to consider to check the representative of data:
 Quality of operators
 Working load on the machine
 Particularity of data observation (e.g., more rainy days
than other years)
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Figure 6.2
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Data set of machine breakdowns
• Example 2: Defective computer chips
 The chip boxes are selected at random from …..
• Points to check on data:
 What is the data type?
 Are the data representative?
 How the randomness of data realized?
• Statistical problem:
 What is the population from which the data are sampled?
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Figure 6.4
Data set of defective
computer chips
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6.2 Data presentation
6.2.1 Bar and Pareto charts
6.2.2 Pie charts
6.2.3 Histograms
6.2.4 Outliers
 An outlier is an observation which is not from the distribution
from which the main body of the sample is collected.
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Figure 6.7 Bar chart of machine breakdowns data set
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Figure 6.9 Pareto chart of customer complaints for Internet company
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Figure 6.12
Pie chart for
machine
breakdowns
data set
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Figure 6.14
Histogram of computer
chips data set
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Figure 6.16 Histograms of metal cylinder diameter data set with
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different bandwidths
Figure 6.18 A histogram with positive skewness
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Figure 6.19 A histogram with negative skewness
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Figure 6.21 Histogram of a data set with a possible outlier
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6.3 Sample statistics
6.3.1 Sample mean
6.3.2 Sample median
6.3.3 Sample trimmed mean
6.3.4 Sample mode
6.3.5 Sample variance
6.3.6 pth Sample quantiles
6.3.7 Boxplots
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Cf. Chebyshev’s inequality:
2
Let E[ X ]   \ and \ Var ( X )   .
Then, P{| X   | c }  1  1/ c 2 \ or
P{| X   | c }  1/ c 2 .
In general, P{| X   |  }   2 /  2 .
Cf. Theorem: the weak law of large numbers
Let X i , i  1, , n be a sequence of i.i.d. random
variables, each having mean  and variance  2 .
Then, for any   0,
lim P{| X   |  }  0
n 
1 n
where \ X   X i
n i 1
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(proof)
E[ X ]   \ and \ Var ( X )   2 / n.
It \ follows \ from \ Chebyshev ' s \ inequality \ that
2
P{| X   |  }  2 .
n
Therefore,
lim P{| X   |  }  0.
n 
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Figure 6.22
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Illustrative data set
Figure 6.23
Relationship between the
sample
mean, median, and trimmed
mean
for positively and negatively
skewed data sets
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Figure 6.20 A histogram for a bimodal distribution
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Figure 6.24
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Boxplot of a data set
Figure 6.30
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Rolling mill process
Figure 6.31
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% scrap data set from rolling mill process
Figure 6.32 Histogram of rolling mill scrap data set
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Figure 6.33
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Boxplot and summary statistics for
rolling mill scrap data set